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QuestionRearrange 2000=1000(1+R100)D2000=1000\left(1+\frac{R}{100}\right)^{D} to find D=log⁡(2)log⁡(1+R100)D=\frac{\log (2)}{\log \left(1+\frac{R}{100}\right)}.

Studdy Solution

STEP 1

Assumptions1. The given expression is 2000=1000(1+R100)2000=1000\left(1+\frac{R}{100}\right)^{} . We need to make $$ the subject of the expression

STEP 2

First, we can simplify the equation by dividing both sides by1000.
20001000=(1+R100)\frac{2000}{1000}=\left(1+\frac{R}{100}\right)^{}

STEP 3

implify the left side of the equation.
2=(1+R100)2=\left(1+\frac{R}{100}\right)^{}

STEP 4

To isolate ,wecantakethenaturallogarithm(log)ofbothsides.Thereasonwedothisisbecausethelogarithmofanumbertoacertainbaseistheexponenttowhichthebasemustberaisedtogetthenumber.Inthiscase,wewanttofindtheexponent, we can take the natural logarithm (log) of both sides. The reason we do this is because the logarithm of a number to a certain base is the exponent to which the base must be raised to get the number. In this case, we want to find the exponent .
log⁡(2)=log⁡((1+R100))\log(2)=\log\left(\left(1+\frac{R}{100}\right)^{}\right)

STEP 5

Using the property of logarithms that log⁡(ab)=blog⁡(a)\log(a^b)=b\log(a), we can bring $$ out of the exponent.
log⁡(2)=log⁡(1+R100)\log(2)=\log\left(1+\frac{R}{100}\right)

STEP 6

Finally, to solve for $$, we can divide both sides of the equation by $\log\left(1+\frac{R}{100}\right)$.
=log⁡(2)log⁡(1+R100)=\frac{\log (2)}{\log \left(1+\frac{R}{100}\right)}So, =log⁡(2)log⁡(1+R100)=\frac{\log (2)}{\log \left(1+\frac{R}{100}\right)} is the solution.

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